Sokhobiddin Akhatkulov

On Critical Circle Homeomorphisms with Infinite Number of Break Points

We prove that a critical circle homeomorphism with infinite number of break points without periodic orbits is conjugated to the linear rotation by a quasisymmetric map if and only if its rotation number is of bounded type. And we also prove that any two adjacent atoms of dynamical partition of a unit circle are comparable.

Multidimensional fixed-point theorems and applications

The purpose of this work is to present the applications of multidimensional fixed point theorems. For this, we prove some multidimensional fixed point theorems and then using these theorems we show the existence and uniqueness of solution of a systems of matrix equations.

A necessary condition for the absolute continuity of invariant measure of circle maps with countably infinite number of break points

Let f be a circle homeomorphism with countably many break points that is, differentiable except in countably many points where the derivatives have a jump. Assuming its rotation number ρ to be irrational, we provide a necessary condition for the absolute continuity of invariant measure with respect to the Lebesgue measure.

An analogue of the prime number, Mertens’ and Meissel’s theorems for closed orbits of the Dyck shift

In this paper, we prove the dynamical analogues of the Prime number, Mertens’ and Meissel’s theorems for the closed orbits of the Dyck shift.

Circle homeomorphism with infinite type of singularities

We prove, that the cross-ratio distortion with respect to a given circle homeomorphism with infinite number of break and finite number of singular points is bounded.

On conjugacies of circle maps with singular points

Let Tf and Tg be P-homeomorphisms of the circle with break point singularities that is, differentiable away from countable many points where the derivative has a jump. Let Tf and Tg have the same irrational rotation number. We describe a formula to find a conjugating map Th between Tf and Tg, i.e., Th○Tf = Tg○Th. Moreover, we provide a sufficient condition for the β - Holder continuity of the conjugating map.

On quasi-symmetry of conjugacies between critical circle maps with break points

We prove that a critical circle homeomorphism with infinite number break points without periodic orbits is conjugated to the linear rotation by a quasi-symmetric map if and only if its rotation number is of bounded type.

On smoothness of conjugations between break equivalent circle maps

Let Tf and Tg be a P-homeomorphisms of the circle with break point singularities that is, differentiable away from countably many points where the derivative has a jump. Let Tf and Tg have the same irrational rotation number. We provide a sufficient and necessary condition for the C1 -smoothness of conjugation map between break equivalent homeomorphisms Tf and Tg.